text: In a suspension of solid particles (e.g. micron-size spheres suspended in oil), an effective viscosity  can be defined in terms of stress and strain components which are averaged over a volume large compared with the distance between the suspended particles, but small with respect to macroscopic dimensions. Such suspensions generally exhibit non-Newtonian behavior. However, for dilute systems in steady flows, the behavior is Newtonian and expressions for  can be derived directly from the particle dynamics. In a very dilute system, with volume fraction , interactions between the suspended particles can be ignored. In such a case one can explicitly calculate the flow field around each particle independently, and combine the results to obtain . For spheres, this results in the Einstein's effective viscosity formula:
  where  is the viscosity of the suspending liquid. The linear dependence on  is a consequence of neglecting interparticle interactions. For dilute systems in general, one expects  to take the form
  where the coefficient  may depend on the particle shape (e.g. spheres, rods, disks). Experimental determination of the precise value of  is difficult, however: even the prediction  for spheres has not been conclusively validated, with various experiments finding values in the range . This deficiency has been attributed to difficulty in controlling experimental conditions.
  In denser suspensions,  acquires a nonlinear dependence on , which indicates the importance of interparticle interactions. Various analytical and semi-empirical schemes exist for capturing this regime. At the most basic level, a term quadratic in  is added to :
  and the coefficient  is fit from experimental data or approximated from the microscopic theory. However, some authors advise caution in applying such simple formulas since non-Newtonian behavior appears in dense suspensions ( for spheres), or in suspensions of elongated or flexible particles.
  There is a distinction between a suspension of solid particles, described above, and an emulsion. The latter is a suspension of tiny droplets, which themselves may exhibit internal circulation. The presence of internal circulation can decrease the observed effective viscosity, and different theoretical or semi-empirical models must be used.